Give an example for a continous function need not be of bounded variation
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x is not an extreme point of CIt is easy to show a function that is not of bounded variation. Consider the function \(f\) defined on the interval \([0,1]\) by:
\[f(x)=\left\{
\begin{array}{ll}
0 & \mbox{if } x=0\\
Introduction on total variation of functions
Recall that a function of bounded variation, also known as a BV-function, is a real-valued function whose total variation is bounded (finite).
Being more formal, the total variation of a real-valued function \(f\), defined on an interval \([a,b] \subset \mathbb{R}\) is the quantity:
\[V_a^b(f) = \sup\limits_{P \in \mathcal{P}} \sum_{i=0}^{n_P-1} \left\vert f(x_{i+1}) – f(x_i) \right\vert\] where the supremum is taken over the set \(\mathcal{P}\) of all partitions of the interval considered.\frac{1}{x} & \mbox{if } x \in (0,1]\\
\end{array}
\right.\]
For \(0 < u < 1\), we have \(V_0^1(f) > V_u^1(f) = \frac{1}{u} – 1\) and taking \(u\) as small as desired we get \(V_0^1(f) = +\infty\). The function \(f\) is not bounded on its domain.
x is not an extreme point of CIt is easy to show a function that is not of bounded variation. Consider the function \(f\) defined on the interval \([0,1]\) by:
\[f(x)=\left\{
\begin{array}{ll}
0 & \mbox{if } x=0\\
Introduction on total variation of functions
Recall that a function of bounded variation, also known as a BV-function, is a real-valued function whose total variation is bounded (finite).
Being more formal, the total variation of a real-valued function \(f\), defined on an interval \([a,b] \subset \mathbb{R}\) is the quantity:
\[V_a^b(f) = \sup\limits_{P \in \mathcal{P}} \sum_{i=0}^{n_P-1} \left\vert f(x_{i+1}) – f(x_i) \right\vert\] where the supremum is taken over the set \(\mathcal{P}\) of all partitions of the interval considered.\frac{1}{x} & \mbox{if } x \in (0,1]\\
\end{array}
\right.\]
For \(0 < u < 1\), we have \(V_0^1(f) > V_u^1(f) = \frac{1}{u} – 1\) and taking \(u\) as small as desired we get \(V_0^1(f) = +\infty\). The function \(f\) is not bounded on its domain.
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